# Questions tagged [calculus-of-variations]

This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.

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### Smoothest logpdf distribution

I want to solve the following optimization $$\min_{f\in C^2(\mathbb{R})}\int_{-\infty}^{\infty}(\frac{d^2}{dx^2}\log(f(x)))^2 dx$$ subject to the constraint that $$\int_{-\infty}^{\infty} f(x)dx = 1$$...
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### Calculus of variations with linear Lagrangian and two optimisations

I am working on a two-part system, that jointly determines two functions $F$ and $G$. $F'(x) = f(x)$ and $G'(x) = g(x)$. $F$ and $G$ are both defined on $[0, 1]$, $F(0) = G(0) = 0$, they are both non-...
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### Regularity for computing the first variation

I am having a trouble understanding the regularity needed to compute the first variation for the Euler-Lagrange equation for the functional $$F(u) = \int f(u) dx$$ Suppose $u:U \to \mathbb{R}$ for ...
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### Quasiconvexity implies rank-one convexity proof

I'm struggling to understand the proof of Proposition 5.3 in Rindler's 'Calculus of Variations'. I cannot follow why $u_j$ is $0$ on the boundary of $Q_n$. The next line states that $u_j$ converges ...
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### Regarding Order of a function.

While going through VI Arnold's MMCM, I came across the following definition on Pg 56. I will present the relevant part here ... and R(h,$\gamma$)=$O$($h^2$), in the sense that for |h|<$\epsilon$ ...
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### Not enough degrees of freedom in Euler Lagrange equation

I have the following minimization problem $$\min_{p,r} \int_0^\infty (r^2 + p^2) dt$$ subject to $$\dot r = - r + \dot p,\quad p(0)=p_0,\quad r(0)=r_0, \quad p(\infty)=0, \quad r(\infty)=0.$$ ...
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### Result check of a functional derivative

For quantum chemistry problems, I need to understand functional derivative. I have begun to learn them a few days ago. So, my skill is very low. In a exercise, I tried to calculate the functional ...
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### Variations of Dawson's function

I am studying Dawson's function : $\displaystyle F : x \mapsto e^{-x^2}\int_0^x e^{t^2} dt$. I would like to prove that $F$ attains a maximum at a certain value $x_0 \in (0,1)$, and is increasing over ...
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### Why does the partial derivative of $y'$ with respect to $y$ vanish? [closed]

I'm following Weinstock's Calculus of Variations. On page 25 it says: We have in this case $$f=\frac{d g}{d x}=\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}y',$$ so that the Euler-...
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### Functional derivative different when constraints are explicitly substituted vs. Lagrange multiplier method

Question (brief) Is the result the same if one 1) takes a functional derivative on a constrained subspace, or 2) takes a functional derivative using the method of Lagrange multipliers to encode ...
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### Minimizing surface area of revolution for a fixed volume of revolution

Suppose I want an open-topped cup to have the capacity to hold a volume $V$ of liquid. I want to find a shape for my cup that minimises its surface area. My attempt I strongly suspect that the shape ...
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### Analytical solution for minimizing a functional involving a double integral

The core of diffusion models in machine learning is to find a denoiser function to approximate the diffusion noise. However, under simple settings, I would like to know if the optimal "denoiser&...
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### In variational calculus, why weight function equals the variation of the primary variable $w=\delta u$

In the book, "An introduction to FEM" by J.Reddy, p32, it is stated that in the weak formulation, the weight function has the meaning of the variation of the dependent variable, i.e. in the ...
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### Extremal of a functional with one variable endpoint

Consider the functional $$J[y] = \int \limits_{0}^{b} {\frac{\sqrt{y'^2 + 1}}{y} \text{d}x}$$ with $y(0) = 0$ and the other endpoint somewhere along the circle $(x-9)^2 + y^2 - 9 = 0$ (call this ...
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### The Variational form of a biharmonic PDE

Suppose $\Omega \subset \mathbb{R}^d$ is a $C^{1,1}$ domain. Consider the biharmonic boundary value problem (BVP): $$\begin{cases} \Delta^2 u = f \\ \nabla u \cdot \nu = g \\ u = u_D \end{cases}$$ ...
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### variational form of the functional

when reading some article, i have some problem to get the force $\mathbf{F}$ related to the energy functional from Hamilton's principle. Given the functional:  E=\frac{1}{2}k_{s}\int\left(\left|\...
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